Poincaré's Ding-an-Sich

Princeton's Graduate College

When I was a graduate student (1958-61), one of the philosophy majors came
to me and told me what quantum mechanics is. I was not happy with his
attitude and asked him whether he knows how to add waves. I asked specifically
what happens when we add two sound waves with different frequencies. He
then told me that this is a mathematical question, and that it does not carry
its own substance. I thought he was hopeless and told him to go away.

If Bertrand Russell (1872-1970) of England had said the same thing to me
about mathematics, I could not have told him to go away. Russel was one
of the great thinkers of the past century and I like his history books.
Russell told Herni Poincaré (1854-1912) that mathematics is only a
tool of logic and nothing else, but Poincaré repeatedly disagreed.

Did Poincaré clearly define what mathematics is to Russell? We do
not know, but we can try to figure out what Poincaré had in mind.
We can do this not necessarily because we are smarter than he was, but
because we are equipped with tons of progresses in physics made since
Poincaré's death in 1912.

Like Einstein, Poincaré was heavily influenced by
Immanuel Kant. One object
or event could appear differently to different observers, but there is one and
only one absolute entity. Kant used the word "Ding-an-Sich" (thing in itself).
From Poincaré's point of view, mathematics could have been his "Ding-an-Sich."

If Russel and Poincaré are both right, nothing is everything. Even without
those two great names, we usually complain that nothing is everything
according to philosophers.

In preparation for this webpage, I made a trip to Paris 2012 in order to have a
photo of myself at Poincaré's grave. He was buried at the Montparnasse Cemetery 1912.

This is his grave, one hundred
years old in 2012. Many people came here and left their used Metro tickets on the stone.

From the top of the Arch of Triumph,
we can see how tall this structure is compared with other buildings in Paris.
Underneath of the golden dome seen in this photo is Napoleon's gasket. The building
is called "Hotel des Invalides" meaning hospital for wounded soldiers.

From the Sartre Square, the
Montparnasse Tower looks like this. The Sartre square is the area with the Cafe
Flores and the Cafe Deux Magots where Sartre used to preach his philosophy.
Click here for the Sartre Square in Paris.

Let us go back to physics. Speaking of the role of mathematics in physics,
Poincaré started with the three-by-three rotation matrix applicable to the three-dimensional space.
He expanded it to four-by-four by augmenting the time variable. He then came up with
five-by-five matrices in order to take into account the space-time translational variables.
This is the way in which he completed the formulation of the
Poincaré group.

Here is what Raymond Streater says about Poincaré.
Go to
his web page on Henri Poincaré. According to him,
Poincaré formulated the Lorentz-covariant space-time symmetry before Einstein
completed special relativity as a new physical theory. Streater is not only an
outstanding physicist but also a very critical history writer. I had two overlapping
years with him in Princeton (1960-62).

Poincaré and Einstein

Einstein's Relativity.
While many people were worrying about transformation properties
of velocities, Einstein was interested in transformation properties of momentum and
energy, and came-up with the concept of four-momentum. In so doing he obtained one
mathematical formula the energy-momentum relation, leading to his celebrated
E = mc2.

By 1904, Lorentz and Poincaré proved the equations of electromagnetism,
with the velocity of light as an invariant constant, are invariant under Lorentz
transformations. Then, the question is why Newton's
equation is not. Here, Einstein had to face a Hegelian problem: how to make
Newton's equation and Maxwell's equations obey the same transformation law.
He solved this problem and came up with a relativistic form Newton's equation.
During this process, Einstein found out the momentum and energy can become a
four-vector transforming like the space-time four-vector.

Hermann Minkowski was born in
Lithuania and studied in Koenigsberg. He was a devoted Kantianist, and a also
a creative mathematician. He was one of Einstein's teachers.
In 1907, he published a paper where he gave a geometrical interpretation
Lorentz transformations. He was particularly interested in expressing
everything in terms of quadratic forms.

Lorentz transformations in the four-dimensional Minkowskian space include
both rotations and squeeze transformations. Rotations are quite familiar
to physicists. The squeeze is one of the standard deformations in engineering,
but this transformation is still strange to physicists.

Let us go back to Kant. It is easy to say that Einstein's relativity is inconsistent
with Kant's Ding-an-Sich which is the absolute coordinate frame. Let us see what Einstein
did more closely. He divided many different observations into two groups (as in the
case of Taoism). He then combined these two into one. Yes, Einstein was a Kantianist
with the Lorentz-covariant energy-momentum relation as his Ding-an-Sich. Einstein's
Ding-an-Sich is different from Kant's Ding-an-Sich. It appears as a mathematical formula.

Click here for an illustration of how Einstein
reached his Ding-an-Sich. He had to go through Taoism and Hegelianism.

This image will illustrate how Kantianism
and Taoism were developed from the same geographical environment. Kantianism is a
product of the geographical condition of
Koenigsberg. Like Venice, Koenigsberg
was the traffic center for the ships navigating in the Baltic Sea. Many people came
there with different ideas. The city had to find common ground to entertain those
people. This is what Kant's Ding-an-Sich is all about.

China was created as a collection of many isolated population pockets with different
backgrounds. They came to one place with different ideas. They divided those ideas
into two opposing groups. This is what Taoism is all about.

Two-party system of American
government was developed in the same way as Taoism was developed in ancient China.

From these illustrations, we now have a clearer picture of what Kantianism is all about. From
Poincaré's point of view, Einstein's Ding-an-Sich is mathematics. The mathematical
expression of Einstein's energy-momentum relation is much more than an instrument of
logic.

Poincaré Group after Einstein

When Einstein formulated his special relativity in 1905 and derived his energy-momentum
relation, he did not take into consideration of the fact that particles have internal
space-time symmetries. Then how can we define these symmetries?

After 1927, we had to worry about this problem in the quantum world. For the Lorentz
group, we have three-rotational degrees of freedom as well as three boost degrees.
For each rotation, there is an angular momentum associated with this transformation.
For boosts, we still are not able to associate dynamical variables since boost
generators are not Hermitian operators.

How about momentum? Indeed, Einstein's four-momentum is associated with space-time
translations in the Minkowskian space. If the momentum variable is to play a role,
we have to resort to the inhomogeneous Lorentz group where the translational degrees
of freedom are augmented to the Lorentz group with the rotation and boost generators.
This inhomogeneous Lorentz group is commonly called the Poincaré group.

If we want to study the internal space-time symmetry of a particle, we should consider
the symmetry for a fixed value of its momentum. This is necessarily a subgroup of
the Lorentz group with three degrees of freedom. It was Eugene Wigner who addressed
this issue in 1939.

In order to discuss this problem, we need many webpages. You may go to
this page for a brief survey of
this field. You will note there that this subject has been my main business since 1973.

Poincaré's Geometry and Topology

Poincaré was quite fond of drawing pictures when he was thinking.
He used the surface of a sphere to describe the polarization of light.
This sphere is known as the Poincaré sphere, and discussed in
every optics textbooks. Like the Eulier angles, this sphere carries
three variables.

The sphere has its radius. If it is allowed to vary, the sphere has
four variables. In this case, did you know that the sphere can be
used for representations of the Lorentz group which Poincaré
formulated? I talked about this aspect in one of my earlir.

Click here for a paper
I presented at the Fedorov Memorial Symposium: International
Conference "Spins and Photonic Beams at Interface," dedicated to the 100th
anniversary of F. I. Fedorov (Minsk, Belarus, 2011). Academian Fedorov
was one on the pioneers in applications of the Lorentz group in optical sciences.

One mathematics for two
different branches of physics. The damped harmomic oscillator and the LCR circuit are two different manifestations
of the second-degree differential equation. Likewise, special relativity
and polarization phyiscs are two different manifestations of the Lorentz
group.

The Lorentz group is applicable to many other branches of optical sciences.
For a list my paper on this subject, go to
this page.

Here again, Poincaré had enough reason to think mathematics is
his Ding-an-Sich.

Poincaré was also interested in circles. He was particularly interested
in under what circumstances we can shrink a circle continuously to a point.

We would not be working against his wishes if we worry about how a
circle can be deformed while preserving its area. It can become an ellipse and
eventually become a straight line.
Click here for the ellipse, and
here for a detailed story. Here again, we are talking about the
Lorentz group formulated first by Poincaré.

Here again, in the spirit of Poincaré, we can worry about
to what extent a hyperbola is the same as an ellipse. I studied this aspect
in this webpage. We know one can
be continuously transformed into the other, but art they the same?

Let us start with a hyperbola which can be written as

x2 - y2 = 1.

We can write this as

(x + y)(x - y ) = 1.

Here, if (x + y) becomes larger, (x - y) has to
becomes smaller. What does this mean?

Let us start with a circle

x2 + y2 = 1,

which can also be written as

(x + y)2 + (x - y)2 = 2.

If (x + y) becomes larger, (x - y) has to
becomes smaller, it is an elliptic deformation of the above equation.
Thus, the circle is the same as the hyperbola to the extent that the
first hypergolic equation leads to an elliptic deformation of the
cricle. Click here
for a story. What physics can you do with this?
Click here.

I am very happy to say that I was able to reach this conclusion based
on analytic geometry I learned during my high-school senior year (1953-54) in
Korea.

Two and Three

The simplest numbers are ZERO and ONE (solid-state devices can handle these two numbers).
TWO is closer to ONE, but is much larger than ONE. How about THREE. This number is
very close to TWO, but is also closer to INFINITY. I have been worrying about this number
for some time, and I have a webpage dedicate
to THREE. I have another webpage on this issue.
My PhD thesis (1961) was on the three-body problem.

However, I am not the first one to see this point. Henri Poincaré was the first
one to recognize the significance of this number. I am encouraged, and I have to do some
more work to understand my own papers before making a meaningful statement about
Poincaré's work on this subject. Please come again.

Conclusion

Poincaré's Ding-an-Sich was mathematics. This is the reason why Bertrand
Russell said Henri Poincaré was the greatest man France had ever produced,
even though he insisted that mathematics does not have its own content. After all,
philosophers are right. Nothing is everything.

Since I have been writing papers for more than fifty years, I am entitled to organize
my publications within some framework. It appears that they could fit into Poincaré's
world. Thus, Henri Poincaré is my Ding-an-Sich.

It is very easy for young physicists to get lost these days. If you think you are lost
in physics, find out your own Ding-an-Sich. If you thinks you are an established
physicist, but you are not sure about whether your work will remain in history, find out
what your Ding-an-Sich is.

Other Poincaré Papers

Wigner's Little Groups dictate
internal space-time symmetries of elementary particles. Wigner's work is based
on the Poincaré group governing both Lorentz transformations and translations.
This group is called the inhomogenous Lorentz group.

Raymond Poincaré was Henri's
cousin. He was the president of France from 1913 to 1920, covering the period
of World War I. I am not able to write a story about him. Perhaps we can
rely on his
Wikipage.

The photo of the Poincaré sphere on this webpage came from
Christian Brosseau's book entitled "Fundamentals of Polarized Light,
A Statistical Optics Approach" (Wiley, New York, 1998). I am
grateful to Professor Brosseau for sending me a copy of this book.